Faculty of Social Sciences Induction Block: Maths

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Faculty of Social Sciences Induction Block
Maths Statistics Lecture 1

• Overview Variables, Constants, Tables Graphs
• Dr Gwilym Pryce

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Aims and Objectives of the Maths Stats
Induction

• Aim to revise basic maths relevant to the
  course.
• Objectives by the end of the Induction Programme
  students should be able to
• Understand the meaning and types of variables and
  constants
• Understand how to graph scale and categorical
  variables
• Be familiar with basic algebraic notation
• Understand the simple mathematical representation
  of relationships, both algebraically and
  graphically
• Understand the basic principles and laws of
  probability
• Outline the main issues surrounding sampling.

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Why do social scientists need to learn about
statistics?

• Theories have to be verified empirically
  otherwise they remain conjectures
• Need for evidenced based practice policy
• medicine
• public health
• economics
• informed decisions better than uninformed
  decisions
• information is complex and needs summarising in a
  way that reflects the underlying data in a
  meaningful way

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Why do we need mathematics?

• Statistics can be represented in a
  non-mathematical way, but some understanding and
  application of maths will help us
• spoken language can be ambiguous varies across
  countries and cultures

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• Different cultures find different things funny
• Different cultures and languages express ideas
  differently
• But mathematical notation is
• unambiguous and concise
• common notation is understood across cultures and
  languages
• Research ideas expressed mathematically can
  easily reach an international audience

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Plan of Maths Stats Induction

• Lecture 1 Variables, Constants, Tables Graphs
• Lecture 2 Algebra and Notation
• Lecture 3 Precise and Approx Relationships
  between variables
• Lecture 4 Probability
• Lecture 5 Inference
• Lecture 6 Hypothesis tests
• Tutorial Samples and populations Validity and
  Reliability

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Plan of Maths Stats Lecture 1 Variables and
Constants

• 1. What is a variable?
• 2. What is a constant?
• 3. Types of variables
• 4. Graphs of single variables
• Why summarise?
• Tables graphs of categorical data
• Tables Graphs of Continuous /
  Quantitative/Scale variables

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1. What is a variable?

• A measurement or quantity that can take on more
  than one value
• E.g. size of planet varies from planet to
  planet
• E.g. weight varies from person to person
• E.g. gender varies from person to person
• E.g. fear of crime varies from person to person
• E.g. income varies from HH to HH
• I.e. values vary across individuals the
  objects described by our data

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• Individuals basic units of a data set whom we
  observe or experiment on in a controlled way
• not necessary persons
• (could be schools, organisations, countries,
  groups, policies, or objects such as cars or
  safety pins)
• Variables information that can vary across the
  individuals we observe
• e.g. age, height, gender, income, exam scores,
  whether signed Nuclear Test Ban Treaty

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2. What is a constant?

• A measurement or quantity that has only one value
  for all the objects described in our data
• Also called a scalar or intercept or
  parameter
• E.g. speed of light in a vacuum constant for all
  light transmissions
• E.g. ratio of diameter to circumf. constant for
  all circles
• E.g. ave. increase in life expectancy constant
  at 1 year pa since 1900

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• Often it is a constant that want to estimate
• we employ statistical techniques to estimate
  parameters or constants that summarise or
  link variables.
• e.g. mean typical value of a variable
  measure of central tendency
• e.g. standard deviation measure of the
  variability of a variable measure of spread
• e.g. correlation coefficient measures the
  correlation between two variables
• e.g. slope coefficients how much y increases
  when x increases

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3. Types of variables

• Numeric values are numbers that can be used in
  calculations.
• String Values are not numeric, and hence not
  used in calculations.
• But can often be coded I.e. transformed into a
  numerical variable
• e.g. If (country Argentina) X 1.
• If (country Brazil) X 2. etc.

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• Scale or quantitative Variables data values are
  numeric values on an interval or ratio scale
• (e.g., age, income). Scale variables must be
  numeric.
• E.g. dimmer switch brightness of light can be
  measured along a continuum from dark to full
  brightness
• Categorical Variables variables that have
  values which fall into two or more discrete
  categories
• E.g. conventional light switch either total
  darkness or full brightness, on or off.
• Male or female, employment category, country of
  origin

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Two types of Ordinal variables

• Ordinal variables Data values represent
  categories with some intrinsic order
• (e.g., low, medium, high strongly agree, agree,
  disagree, strongly disagree).
• Ordinal variables can be either string
  (alphanumeric) or numeric values that represent
  distinct categories (e.g., 1low, 2medium,
  3high).

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Ordinal variables

• Values fall within discrete but ordered
  categories
• I.e. the sequence of categories has meaning
• e.g. education categories
• 1 primary
• 2 secondary
• 3 college
• 4 university undergraduate
• 5 university postgraduate masters
• 6 university postgraduate phd
• e.g. 1 Very poor, 2 poor, 3good, 4very good

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Nominal variables

• Nominal Variables Data values represent
  categories with no intrinsic order
• sequence of categories is arbitary -- ordering
  has no meaning in and of itself
• e.g. country of origin Wales, Scotland, Germany
• e.g. make of car Ford, Vauxhall
• e.g. job category
• e.g. company division
• Nominal variables can be either string
  (alphanumeric) or numeric values that represent
  distinct categories (e.g., 1Male, 2Female).

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4. Graphs of Variables

• Why summarise?
• Tables graphs of categorical data
• Tables Graphs of Continuous /
  Quantitative/Scale variables

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Why Summarise?

• Small data sets can be presented in their
  entirety
• e.g. if only have 10 observations and 3
  variables, can list all data
• but even then we might want to know what is the
  typical value of a variable
• Large data sets require summary
• Lots of information can be confusing,
  particularly if numerical
• most of us need headline figures or stylised
  facts to be able to absorb information.

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• Graphical summaries
• allow us to visualise the distribution of data
  across different values or categories
• how many (or what proportion) of cases fall
  within certain categories or ranges of values?
• Summary statistics
• describe the distribution of a single variable

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Tables of Categorical Data

• Categories are listed either in columns or rows
  (respecting order if ordinal)
• Count or of cases in each category listed
• If number of categories is large, may be useful
  to group categories together
• e.g. Country of origin ---gt collapse to
  continents
• Good tables
• give clear messages tell a story
• too much info in a table defeats its purpose
• Source always given

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Graphs of Categorical Data

• Pie Charts
• If all the categories sum to a meaningful total,
  then you can use a pie chart
• Pie charts emphasise the differences in
  proportions between categories
• OK for a single snapshot, but not very good for
  showing trends
• would need to have a separate pie chart for each
  year

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Whats missing?
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• Bar Charts
• can show either or count
• not very good for showing trends in more than one
  category

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Beware of scaling...
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Beware of small print...
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Tabulating and Graphing Scale Data

• Scale or quantitative data usually a measurement
  of size or quantity
• not meaningful to report or count unless break
  into categories ( then it becomes categorical
  data!)
• e.g. income
• Tables of raw data not much use unless only a few
  values...

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How tabulate 129,000 observations?
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• What are we interested in when describing the
  income data?
• Is income evenly spread?
• Or are most people rich?
• Or are most people poor?
• Or are most reasonably well off?
• This are all questions about the variables
  Distribution
• We can represent the whole data set with one
  picture...

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